Chomsky normal form

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In formal language theory, a context-free grammar G is said to be in Chomsky normal form (discovered by Noam Chomsky)[1] if all of its production rules are of the form:[2]

ABC,   or
Aa,   or
S → ε,

where A, B, and C are nonterminal symbols, a is a terminal symbol (a symbol that represents a constant value), S is the start symbol, and ε denotes the empty string. Also, neither B nor C may be the start symbol, and the third production rule can only appear if ε is in L(G), namely, the language produced by the context-free grammar G.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form and has a size no larger than the square of the original grammar's size.

Converting a grammar to Chomsky normal form[edit]

To convert a grammar to Chomsky normal form, a sequence of simple transformations is applied in a certain order; this is described in most textbooks on automata theory.[3][4][5][6] The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).[7][note 1] Each of the following transformations establishes one of the properties required for Chomsky normal form.

START: Eliminate the start symbol from right-hand sides[edit]

Introduce a new start symbol S0, and a new rule

S0S,

where S is the previous start symbol. This doesn't change the grammar's produced language, and S0 won't occur on any rule's right-hand side.

TERM: Eliminate rules with nonsolitary terminals[edit]

To eliminate each rule

AX1 ... a ... Xn

with a terminal symbol a being not the only symbol on the right-hand side, introduce, for every such terminal, a new nonterminal symbol Na, and a new rule

Naa.

Change every rule

AX1 ... a ... Xn

to

AX1 ... Na ... Xn.

If several terminal symbols occur on the right-hand side, simultaneously replace each of them by its associated nonterminal symbol. This doesn't change the grammar's produced language.[2]:Thm.4.5,p.92

BIN: Eliminate right-hand sides with more than 2 nonterminals[edit]

Replace each rule

AX1 X2 ... Xn

with more than 2 nonterminals X1,...,Xn by rules

AX1 A1,
A1X2 A2,
... ,
An-2Xn-1 Xn,

where Ai are new nonterminal symbols. Again, this doesn't change the grammar's produced language.[2]:Thm.4.5,p.93

DEL: Eliminate ε-rules[edit]

An ε-rule is a rules of the form

A → ε,

where A is not the grammar's start symbol.

To eliminate all rules of this form, first determine the set of all nonterminals that derive ε. Hopcroft and Ullman (1979) call such nonterminals nullable, and compute them as follows:

  • If a rule A → ε exists, then A is nullable.
  • If a rule AX1 ... Xn exists, and each Xi is nullable, then A is nullable, too.

Obtain an intermediate grammar by replacing each rule

AX1 ... Xn

by all versions with some nullable Xi omitted. By deleting in this grammar each ε-rule, unless its left-hand side is the start symbol, the transformed grammar is obtained.[2]:Thm.4.3,p.90

For example, in the following grammar, with start symbol S0,

S0AbB | C
BAA | AC
Cb | c
Aa | ε

the nonterminal A, and hence also B, is nullable, while neither C nor S0 is. Hence the following intermediate grammar is obtained:[note 2]

S0AbB | AbB | AbB | AbB   |   C
BAA | AA | AA | AεA   |   AC | AC
Cb | c
Aa | ε

In this grammar, all ε-rules have been "inlined at the call site".[note 3] In the next step, they can hence be deleted, yielding the grammar:

S0AbB | Ab | bB | b   |   C
BAA | A   |   AC | C
Cb | c
Aa

This grammar produces the same language as the original example grammar, viz. {ab,aba,abaa,abab,abac,abb,abc,b,bab,bac,bb,bc,c}, but apparently has no ε-rules.

UNIT: Eliminate unit rules[edit]

A unit rule is a rule of the form

AB,

where A, B are nonterminal symbols. To remove it, for each rule

BX1 ... Xn,

where X1 ... Xn is a string of nonterminals and terminals, add rule

AX1 ... Xn

unless this is a unit rule which has already been removed.

Order of transformations[edit]

When choosing the order in which the above transformations are to be applied, it has to be considered that some transformations may destroy the result achieved by other ones. For example, START will re-introduce a unit rule if it is applied after UNIT. The table shows which orderings are admitted.

Moreover, the worst-case bloat in grammar size[note 4] depends on the transformation order. Using |G| to denote the size of the original grammar G, the size blow-up in the worst case may range from |G|2 to 22 |G|, depending on the transformation algorithm used.[7]:Tab.3,p.7 The blow-up in grammar size depends on the order between DEL and BIN. It may be exponential when DEL is done first, but is linear otherwise. UNIT can incur a quadratic blow-up in the size of the grammar.[7]:p.5 The orderings START,TERM,BIN,DEL,UNIT and START,BIN,DEL,UNIT,TERM lead to the least (i.e. quadratic) blow-up.

Example[edit]

Abstract syntax tree of the arithmetic expression "a^2+4*b" wrt. the example grammar (top) and its Chomsky normal form (bottom)

The following grammar, with start symbol Expr, describes a simplified version of the set of all syntactical valid arithmetic expressions in imperative programming languages like C or Algol60. Both number and variable are considered terminal symbols here for simplicity, since in a compiler front-end their internal structure is usually not considered by the parser. The terminal symbol "^" denoted exponentiation in Algol60.

Expr Term | Expr AddOp Term | AddOp Term
Term Factor | Term MulOp Factor
Factor Primary | Factor ^ Primary
Primary number | variable | ( Expr )
AddOp → + | −
MulOp → * | /

In step "START" of the above conversion algorithm, just a rule S0Expr is added to the grammar. After step "TERM", the grammar looks like this:

S0 Expr
Expr Term | Expr AddOp Term | AddOp Term
Term Factor | Term MulOp Factor
Factor Primary | Factor PowOp Primary
Primary number | variable | Open Expr Close
AddOp → + | −
MulOp → * | /
PowOp → ^
Open → (
Close → )

After step "BIN", the following grammar is obtained:

S0 Expr
Expr Term | Expr AddOp_Term | AddOp Term
Term Factor | Term MulOp_Factor
Factor Primary | Factor PowOp_Primary
Primary number | variable | Open Expr_Close
AddOp → + | −
MulOp → * | /
PowOp → ^
Open → (
Close → )
AddOp_Term AddOp Term
MulOp_Factor MulOp Factor
PowOp_Primary PowOp Primary
Expr_Close Expr Close

Since there are no ε-rules, step "DEL" doesn't change the grammar. After step "UNIT", the following grammar is obtained, which is in Chomsky normal form:

S0 number | variable | Open Expr_Close | Factor PowOp_Primary | Term MulOp_Factor | Expr AddOp_Term | AddOp Term
Expr number | variable | Open Expr_Close | Factor PowOp_Primary | Term MulOp_Factor | Expr AddOp_Term | AddOp Term
Term number | variable | Open Expr_Close | Factor PowOp_Primary | Term MulOp_Factor
Factor number | variable | Open Expr_Close | Factor PowOp_Primary
Primary number | variable | Open Expr_Close
AddOp → + | −
MulOp → * | /
PowOp → ^
Open → (
Close → )
AddOp_Term AddOp Term
MulOp_Factor MulOp Factor
PowOp_Primary PowOp Primary
Expr_Close Expr Close

The Na introduced in step "TERM" are PowOp, Open, and Close. The Ai introduced in step "BIN" are AddOp_Term, MulOp_Factor, PowOp_Primary, and Expr_Close.

Alternative definition[edit]

Chomsky reduced form[edit]

Another way[8] to define the Chomsky normal form is:

A formal grammar is in Chomsky reduced form if all of its production rules are of the form:

A \rightarrow\, BC or
A \rightarrow\, a,

where A, B and C are nonterminal symbols, and a is a terminal symbol. When using this definition, B or C may be the start symbol. Only those context-free grammars which do not generate the empty string can be transformed into Chomsky reduced form.

Floyd normal form[edit]

In a paper where he proposed a term Backus-Naur Form (BNF), Donald E. Knuth implied a BNF "syntax in which all definitions have such a form may be said to be in "Floyd Normal Form","

\langle A \rangle ::= \, \langle B \rangle \mid \langle C \rangle or
\langle A \rangle ::= \, \langle B \rangle \langle C \rangle or
\langle A \rangle ::=\, a,

where \langle A \rangle, \langle B \rangle and \langle C \rangle are nonterminal symbols, and a is a terminal symbol, because Robert W. Floyd found any BNF syntax can be converted to the above one in 1961.[9] But he withdrew this term, "since doubtless many people have independently used this simple fact in their own work, and the point is only incidental to the main considerations of Floyd's note."[9]

Application[edit]

Besides its theoretical significance, CNF conversion is used in some algorithms as a preprocessing step, e.g., the CYK algorithm, a bottom-up parsing for context-free grammars, and its variant probabilistic CKY.[10]

See also[edit]

Notes[edit]

  1. ^ For example, Hopcroft, Ullman (1979) merged TERM and BIN into a single transformation.
  2. ^ indicating a kept and omitted nonterminal N by N and N, respectively
  3. ^ If the grammar had a rule S0 → ε, it could not be "inlined", since it had no "call sites". Therefore it couldn't be deleted in the next step.
  4. ^ i.e. written length, measured in symbols

References[edit]

  1. ^ Chomsky, Noam (1959). "On Certain Formal Properties of Grammars" (PDF). Information and Control 2: 137–167. 
  2. ^ a b c d Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages and Computation. Reading, Massachusetts: Addison-Wesley Publishing. ISBN 0-201-02988-X.  Section 4.5, Theorem 4.5, p.92; see also the bibliographic notes on p.106
  3. ^ Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages and Computation. Reading, Massachusetts: Addison-Wesley Publishing. ISBN 0-201-02988-X.  Section 4.4-4.5, p.87-92
  4. ^ Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2006). Introduction to Automata Theory, Languages, and Computation (3rd ed.). Addison-Wesley. ISBN 0-321-45536-3.  Section 7.1.5, p.272
  5. ^ Elaine Rich (2007). Automata, Computability, and Complexity: Theory and Applications (1st ed.). Prentice-Hall. ISBN 978-0132288064. [page needed]
  6. ^ Ingo Wegener (1993). Theoretische Informatik - Eine algorithmenorientierte Einführung. Leitfäden und Mongraphien der Informatik. Stuttgart: B.G. Teubner. ISBN 978-3-519-02123-0.  Section 6.2 "Die Chomsky-Normalform für kontextfreie Grammatiken", p.149-152
  7. ^ a b c Lange, Martin; Leiß, Hans (2009). "To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm" (PDF). Informatica Didactica 8. 
  8. ^ Hopcroft, Ullman (1979);[page needed] Hopcroft et al. (2006)[page needed]
  9. ^ a b Knuth, Donald E. 1964. Backus Normal Form vs. Backus Naur Form. Communications of the ACM, 7(12): 735–736, December.
  10. ^ Jurafsky, Daniel; Martin, James H. (2008). Speech and Language Processing (2nd ed.). Pearson Prentice Hall. p. 465. ISBN 978-0-13-187321-6. 

Further reading[edit]

  • Cole, Richard. Converting CFGs to CNF (Chomsky Normal Form), October 17, 2007. (pdf) — uses the order TERM, BIN, START, DEL, UNIT.
  • John Martin (2003). Introduction to Languages and the Theory of Computation. McGraw Hill. ISBN 0-07-232200-4.  (Pages 237–240 of section 6.6: simplified forms and normal forms.)
  • Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X.  (Pages 98–101 of section 2.1: context-free grammars. Page 156.)
  • Sipser, Michael. Introduction to the Theory of Computation, 2nd edition.